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a(n) is the number of terms in the expansion of (x+y-z)*(x^2+y^2-z^2)*(x^3+y^3-z^3)*...*(x^n+y^n-z^n).
+10
3
3, 9, 22, 48, 102, 182, 328, 566, 910, 1396, 2025, 2882, 3976, 5304, 7002, 9071, 11475, 14444, 17886, 21896, 26531, 31880, 37947, 44899, 52657, 61500, 71406, 82383, 94592, 108097, 123017, 139401, 157439, 177134, 198634, 221962, 247378, 274767, 304483, 336533, 371083, 408168, 447944, 490614, 536208
MAPLE
P:= 1;
for n from 1 to 90 do
P:= expand(P*(x^n+y^n-z^n));
A[n]:= nops(P);
od:
MATHEMATICA
Table[Length[Expand[Times@@Table[x^n+y^n-z^n, {n, i}]]], {i, 50}] (* Harvey P. Dale, Oct 02 2018 *)
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 06 2003
a(n) is the number of terms in the expansion of (x-y)*(x^4-y^4)*(x^9-y^9)*...*(x^(n^2)-y^(n^2)).
+10
2
2, 4, 8, 16, 24, 40, 68, 103, 162, 236, 344, 453, 612, 790, 994, 1229, 1432, 1782, 2134, 2517, 2968, 3460, 3974, 4543, 5160, 5822, 6546, 7347, 8184, 9080, 10058, 11075, 12166, 13316, 14536, 15837, 17202, 18654, 20156, 21765, 23450, 25212, 27074, 29001, 31032, 33158, 35370, 37679, 40070, 42578
COMMENTS
In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=1..n} (1-x^(k^2)).
MATHEMATICA
a[n_] := Length@ ExpandAll@ Product[(1 - x^(k^2)), {k, n}]; Array[f, 40]
PROG
(PARI) a(n)=my(P=prod(k=1, n, 1-'x^k^2)); sum(i=0, poldegree(P), polcoeff(P, i)!=0) \\ Charles R Greathouse IV, May 10 2013
a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)*(x^3-y^3)*(x^5-y^5)*...*(x^p_i-y^p_i), where p_i is the i-th prime.
+10
1
2, 4, 6, 8, 10, 20, 22, 36, 42, 66, 90, 110, 142, 184, 232, 284, 342, 400, 458, 532, 604, 678, 756, 838, 928, 1026, 1126, 1230, 1336, 1446, 1558, 1686, 1816, 1954, 2092, 2242, 2392, 2550, 2712, 2880, 3052, 3232, 3412, 3604, 3796, 3994, 4192, 4404, 4626, 4854, 5082
COMMENTS
In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^p_i), where p_i is the i-th prime and p_0 = 1.
Offset is 1 to keep it parallel to other like sequences.
MATHEMATICA
f[n_] := Length@ ExpandAll[(1 - x) Product[(1 - x^Prime[k]), {k, n}]]; Array[f, 51, 0]
a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)*(x^6-y^6)*(x^10-y^10)*...*(x^T_i-y^T_i), where T_i is the i-th triangular number.
+10
0
2, 4, 8, 15, 28, 41, 66, 92, 132, 175, 232, 287, 360, 475, 570, 727, 852, 1009, 1220, 1397, 1646, 1891, 2154, 2441, 2772, 3121, 3508, 3891, 4334, 4791, 5282, 5797, 6376, 6983, 7618, 8285, 8984, 9713, 10500, 11319, 12182, 13093, 14028, 15023, 16064, 17157, 18276, 19447, 20680, 21953
COMMENTS
In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^T_i), where G_i is the i-th triangular number.
MATHEMATICA
f[n_] := Length@ ExpandAll@ Product[1 - x^(k (k + 1)/2), {k, n}]; Array[f, 50]
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